One use of relation liftings is to determine coalgebraic notions of
bisimulation:
[\(L\) -Bisimulation [1]}]
Let \(L\)  be a relation lifting for a functor \(T:\mathbf {Set} \rightarrow \mathbf {Set} \) , and let
\(( X,\xi )\) , \(( Y,\zeta )\)  be \(T\) -coalgebras. A relation
\(S \subseteq X\times Y\)  is an \(L\) -simulation if for all
\(x\in X\)  and \(y\in Y\) ,
\(x\mathbin {S}y\quad \text{implies}\quad \xi (x) \mathbin {LS} \zeta (y).\) 
